Non-Linear Trans-Planckian Corrections of Spectra due to the Non-trivial Initial States
aa r X i v : . [ h e p - t h ] O c t Non-Linear Trans-Planckian Corrections of Spectradue to the Non-trivial Initial States
E. Yusofi ∗ and M. Mohsenzadeh † Department of Physics, Science and Research Branch, Islamic Azad University, Tehran, Iran Department of Physics, Qom Branch, Islamic Azad University, Qom, Iran (Dated: October 2, 2018)Recent Planck results motivated us to use non-Bunch-Davies vacuum. In this paper, we usethe excited-de Sitter mode as non-linear initial states during inflation to calculate the correctedspectra of the initial fluctuations of the scalar field. First, we consider the field in de Sitter space-time as background field and for the non-Bunch-Davies mode, we use the perturbation theory tothe second order approximation. Also, unlike conventional renormalization method, we offer deSitter space-time as the background instead Minkowski space-time. This approach preserve thesymmetry of curved space-time and stimulate us to use excited mode. By taking into account thisalternative mode and the effects of trans-Planckian physics, we calculate the power spectrum instandard approach and Danielsson argument. The calculated power spectrum with this method isfinite, corrections of it is non-linear, and in de Sitter limit corrections reduce to linear form thatobtained from several previous conventional methods.
PACS numbers: 98.80.Cq , 04.62.+v
I. INTRODUCTION
As a causal manner, inflation scenario can be explaindensity perturbations originating in areas outside thehorizon in the very early universe[1, 2]. These perturba-tions actually planted the seeds of the current observablestructure in the universe. During inflation, the physi-cal size of any perturbation mode has grown faster thanthe horizon, cross of it and freezes. After the end ofinflation, these super-horizon perturbed mode re-enterinto the horizon and along with the expansion, form intogalaxies and clusters under the gravitational force[3, 4].Naturally, this causal relations in the sub-horizon radiusallows us to delineate their initial amplitude by consider-ing that perturbations start from quantum fluctuationsof their vacuum.The measurable radiation from the Big Bang is cosmicmicrowave background radiation(CMBR), that providesa snapshot of the early universe and our main notionof the early universe has improved significantly over thesurvey of it. Most of the inflationary models show ananisotropy in the temperature in the CMBR map [5] andit could be indicate the quantum origin of the universeand come from the primordial quantum fluctuations ofthe fields in the initial vacuum state. It may also be re-lated to effects of ”trans-Planckian physics” beyond thePlanck scale [6, 7].Firstly, the trans-Planckian physics effects in inflationwas introduced in [8]. Dispersion relations and their con-cepts in various form have been greatly studied in [9–11].Another way to discuss the trans-Planckian physics em-anated from the quantum gravity and Non-commutative ∗ Electronic address: e.yusofi@iauamol.ac.ir † Electronic address: [email protected]
Geometry of space-time, such as string theory[12–14]. Inthe another notable attempt, Danielsson focused on thechoice of vacuum and he has used α -vacuum state in theinflationary background [6].By consideration of a cutoff length Λ − , the conventionalpredictions of inflation get corrected due to the finite ef-fects of the expansion of the universe [6]. These modi-fications can be expanded as a series in H/ Λ. The or-der of the correction to the power spectrum obtained byDanielsson [6] was H/ Λ. In [15] using the method of ef-fective field theory the authors found that the correctionwas ( H/ Λ) , and when the mode are initially created byadiabatic vacuum state, the authors found that the cor-rection of power spectrum was ( H/ Λ) [16]. Some otherworks has been performed in [17–22]. The complete anal-ysis of these different orders of correction has discussedin [23, 24].Cosmic inflation is described by nearly de Sitter(dS)space-time. In flat space-time, there exists an unique andwell-defined vacuum state, but in curved space-time, themeaning of vacuum is not very clear and exists ambigu-ity in the choice of vacuum[25]. If we consider universeas exact dS space during inflation, there exists a con-crete set of vacuum states invariant under the symmetrygroup of the dS space-time. However, as we know, aninflating universe is not exact dS space-time but it maybe dS space-time in the first approximation [1]. Since,very early universe stood in the high energy area, it mayseem more logical using of higher order perturbations.Therefore, in this work, we exploit of perturbation theoryand consider non-linear excited- dS mode instead linearBunch-Davies(BD) mode as the fundamental mode dur-ing inflation [26, 27]. The main point of this paper is thatthe order of corrections to the power spectrum changes ifwe consider non-linear initial vacuum mode. To achievethis goal, we use the excited version of α -vacuum and wegeneralize Danielsson work in the context of α -vacuum[6].The layout of paper is as follows: In Sec. 2., we brieflyrecall the definition of standard power spectrum and wereview the calculation of standard power spectrum in BDvacuum mode and α -vacuum with trans-Planckian ef-fects. Sec. 3. is the main work of this paper. In 3.1. somemotivations for our offered vacuum mode is introduced.In 3.2. and 3.3., we calculate the corrected power spec-trum with excited-dS vacuum and excited- α − vacuum .Conclusions is given in the final section. II. STANDARD APPROACH FORCALCULATION OF POWER SPECTRUM
The following metric is used to describe the universeduring the inflation: ds = dt − a ( t ) d x = a ( η ) ( dη − d x ) , (1)where for dS space, the scale factor is given by a ( t ) =exp( Ht ) and a ( η ) = − Hη . η is the conformal time and H is the Hubble constant. There are some models ofinflation but the popular and simple one is the singlefield inflation in which a minimally coupled scalar field(inflaton) is considered in inflating background: S = 12 Z d x √− g (cid:16) R − ( ∇ φ ) − m φ (cid:17) , (2)where 8 πG = ~ = 1. The corresponding inflaton fieldequation in Fourier space is given by: φ ′′ k − η φ ′ k + ( k + a m ) φ k = 0 , (3)where prim is the derivative with respect to conformaltime η . For the massless case, with the rescaling of υ k = aφ k , equation (3) becomes υ ′′ k + ( k − ¨ aa ) υ k = 0 . (4)The general solutions of this equation can be written as[15, 28]: υ k = A k H (1) µ ( | kη | ) + B k H (2) µ ( | kη | ) , (5)where H (1 , µ are the Hankel functions of the first andsecond kind, respectively [15]. For the large values of | kη | the above Hankel functions have the expansion to thesecond order approximation in the following form[28, 40], H µ ( | kη | ) ≈ s π | kη | (cid:2) − i ν − kη + (cid:13) ( 1 kη ) (cid:3) × exp [ − ikη ] . (6)Note that for far past we consider | kη | = − kη . A. Power Spectrum with Exact dS Vacuum
Consider the dS limit (H =constant), and¨ aa = 2 η , (7)In a pure dS background, we therefore wish to solve themode equation υ ′′ k + ( k − η ) υ k = 0 . (8)and the exact solution of (8) becomes [2], υ k = A k √ k (1 − ikη ) e − ikη + B k √ k (1 + ikη ) e ikη , (9)where A k and B k are Bogoliubov coefficients. In general,this set of vacua (labelled by α ) is used and written by α -vacuum. However, the free parameters A k and B k canbe fixed to unique values by considering the quantiza-tion condition i ( υ ∗ k ´ υ k − υ ∗ k ´ υ k ) = 1 together with the sub-horizon limit | kη | ≫
1, and leads to the unique Bunch-Davies(BD) vacuum [29] by setting B k = 0 and A k = 1in (9) υ BDk = 1 √ k (1 − ikη ) e − ikη . (10)For any given mode υ k , the two-point function in Hilbertspace is defined by: h φ i = 1(2 π ) Z | υ k | a d k. (11)Then from (10) and (11) one can write: h φ i = 1(2 π ) Z d k [ 12 ka + H k ] . (12)The usual contribution from vacuum fluctuations inMinkowski space-time, i.e. the first term, is divergent.This infinity can be eliminated after the renormalization[1]. One of the simple methods to automatic renormal-ization device in curved space-time, is discussed in [30–36]. Then the renormalized power spectrum for the scalarfield fluctuations is calculated as [1, 37]: P φ ( k ) = ( H π ) . (13)Since during the inflationary era, background space-timeis considered curved, it is better to be a general solu-tion for the wave equation (4), include both positive andnegative norm. The α -vacuum is obtained from (9) ingeneral case, in which B k = 0 and A k = 0. In section(2.3) of [6], Danielsson considered, f k = A k √ k (1 − ikη ) e − ikη + B k √ k (1 + ikη ) e ikη , (14)and g k = r k A k e − ikη − r k B k e ikη , (15)and used the following condition, | A k | − | B k | = 1 , (16)after some straightforward calculation, he obtained thecoefficients as follows: B k = α k A k e − ikη , | A k | = 11 − | α k | , (17)where α k = i kη + i , (18)where η = − (Λ /Hk ) has a finite value. According to[6], we can define a finite η in a way that the physicalmomentum corresponding to k is given by some fixedPlanck scale say as Λ. Similarly, for this general modefunction obtained: h φ i = 1(2 π ) Z d k (cid:20) ( 12 ka + H k ) − H k sin( 2Λ H ) (cid:21) , (19)where (Λ /H ) ≫ k = ap , and p = Λ, af-ter doing some easy calculations, the power spectrum forscalar field fluctuations in this case is obtained as [6, 39] P φ = ( H π ) (cid:16) − H Λ sin( 2Λ H ) (cid:17) . (20)which is a scale-dependent power spectrum and correc-tions are of order H/ Λ. We will generalize the abovemethod to our proposed excited mode.
III. DEPARTURE FROM BUNCH-DAVIESMODE TO THE EXCITED-DE SITTER MODE
In this section, we will introduce the excited-dS modeas a fundamental mode function during inflation. But theimportant question is: why excited-dS mode? Becausewe do not know anything about the physics of the beforeinflation at very early universe, but we know that cosmicinflation can be described by nearly dS space-time. So,any primary excited dS mode can be considered as a goodand acceptable mode for initial state. Of course, this canbe a first step towards to create of the non-linear sourcesof primordial fluctuations for generation anisotropy inCMBR.For the QFT in flat space-time, the vacuum expectationvalue of the energy-momentum tensor becomes infinite which is removed by the normal ordering. However incurved space-time, following trick is used (equ. (4.5) in[25]), h Ω | : T µν : | Ω i = h Ω | T µν | Ω i − h | T µν | i , (21)where | Ω i is the vacuum state in curved space-timewhereas | i stands for the vacuum state of Minkowskianflat background. One can interpreted the mines sign atthe above equation as the effect of the background so-lutions. Note that the symmetry of the curved space-time vividly breaks in this renormalization scheme, be-cause the background solution is not the solutions ofthe wave equation in the curved space-time. Indeed, inthis renormalization procedure the vacuum is defined inglobal curved space-time while the singularities are re-moved in local flat space-time. But, if such divergencesare removed by the quantities which are defined in thecurved space-time, the symmetry would be returned totheory. In this case, the background solutions are thesolutions of the wave equation in the curved space-time[26, 38]. By consideration this new scheme of renormal-ization; theory preserve symmetry of curved space-time.In addition, By having this new mode which certainlycan not be dS mode, we can obtain energy-momentumtensor without any additional handmade cutoff.Now let us, in the first approximation, we consider ex-act dS mode function as the background(BG) solutionand the excited- dS mode as the fundamental solutionsof the curved space-time. This excited-dS solution mustbe asymptotically approaches to dS background. Suchan approximate mode might be obtained by expandingthe Hankel function in (5) up to its third term [26, 28, 40]and then one can write: u exck ≃ √ k (cid:16) − ikη −
12 ( ikη ) (cid:17) e − ikη . (22)Noting that according to the proposed ansatz, the thirdterm in mode (22), is an approximate term that is addeddue to the expansion of the Hankel function according to(6). In the large value of | kη | , we can consider ( kη ) → u BGk = u BDk = 1 √ k (1 − ikη ) e − ikη . (23)We call solution (22) as excited- dS mode [45]. In [26],for the first time, we used this excited solution with theauxiliary fields to calculate the finite and renormalizedpower spectrum of primordial fluctuations. Also, in theour recent work [27], we considered the Planck results(2013) for scalar spectral index of inflation [43], and weshowed that the µ must be greater than the 3/2 and thisimportant result stimulate us departure from linear BDmode to the non-linear excited modes.In addition to the above motivations, corrections ob-tained from previous conventional methods for powerspectrum is typically of the order of 1, 2 or higher. Soit is useful for us to extend the mode up to non-linearorder of its parameters. This non-linearity of our newvacuum mode appears in the conformal time variable η . In this paper, we pursue this topic with Danielssonapproach [6] and in the [26, 27] we have investigated itutilizes other conventional methods. It is also expectedthat the primordial non-Gaussianity of the CMB comefrom various non-linear sources during the cosmic evo-lution. Non-linear term in our excited mode in the ini-tial vacuum may leave non-Gaussian traces in the CMB[41, 44]( This issue will studied in preparing work). Onthe other hand, this excited mode could be more com-plete solution of the general wave equations (4) for thegeneral curved space-time during inflation, whereas BDmode is a specific solution for a specific dS space-time. IV. POWER SPECTRUM WITH EXCITED-DSMODE AND EXCITED- α -VACUUMA. Calculation with excited-dS vacuum By inserting mode function (22) in (11) and doing somestraightforward calculations, one obtains: h φ i = 1(2 π ) Z d k [ 12 ka + H k + a H k ]The first term is the usual contribution from vacuumfluctuations in ds space-time that can be eliminated afterthe renormalization. Then h φ i = 12 π Z dkk ( H + a H k ) . (24)The power spectrum is given by P φ ( k ) = ( H π ) (cid:16) H
2Λ ) (cid:17) , (25)which is scale-dependent and the correction is of order( H/ Λ) , where H is the Hubble parameter during in-flation and Λ is the Planck energy scale. Note that in[15, 42] with effective field theory approach, similar or-der of correction has been obtained. B. Calculation with Excited- α -vacuum Inflation starts in approximate-dS space-time. Basi-cally in this high energy area of very early universe withvarying H , finding a proper mode is difficult. We offerthe excited-dS solution (22) as the fundamental modeduring inflation that asymptotically approaches to dSbackground. Actually, the excited-dS mode (22) can beconsidered as a nearest solution to dS mode. So accord-ing to (14), the general solution of the equation of motionin this approximate-dS space-time, include positive andnegative frequency can be given by, υ k = M k √ k (cid:16) − ikη −
12 ( ikη ) (cid:17) e − ikη + N k √ k (cid:16) ikη −
12 ( − ikη ) (cid:17) e ikη , (26)and call it as excited- α - vacuum . Note that, the modefunctions (14) is the general exact solution of equation(4)for pure dS space-time, and similarly we consider mode(25) as the general approximate solution of equation(4)for approximate-dS space-time. Since this vacuum isof order ( kη ) , for g k we offer two different choices oforders ( kη ) and ( kη ) .
1. First choice of g k Similar to Danielsson work, If we consider the g (1) k , cor-responding to the first derivative or conjugate momentumof υ k we will have, g (1) k = r k M k (1 − ikη ) e − ikη − r k N k (1 + ikη ) e ikη , (27)We follow the section (2.3) in [6] and we obtain for (25)and (26), N k = − γ k M k e − ikη , | M k | = 11 − | γ k | , (28)where γ k = 14( kη ) + 4 ikη + 1 , (29)If we ignore the terms of higher than second order, weobtain for η = − (Λ /Hk ), the corrected power spectrumas, P (1) φ ( k ) = ( H π ) (cid:16) − i H Λ ) cos( 2Λ H ) (cid:17) , (30)Here, the correction in (29) is of order ( H/ Λ) .
2. Second choice of g k In the other hand, conventionally, the vacuum is chosenby requiring that the mode functions υ k reduce to theMinkowski ones in the limit η → −∞ . So, similar toDanielsson work [6], and as the second choice, we consider g (2) k in flat space-time as follows, g (2) k = r k A k e − ikη − r k B k e ikη , (31)and for (25) and (30) we again obtain N k and M k sim-ilar to (27) but for γ k we have, γ k = 2 i ( kη ) + 14( kη ) + 2 i ( kη ) + 1 , (32)and the power spectrum is given as follows, P (2) φ ( k ) = ( H π ) ( 11 − ( H ) ) (cid:16) H
2Λ ) − H Λ sin( 2Λ H ) (cid:17) , (33)If we use the following Taylor expansion for x = H ≪ X x n = 11 − x , (34)and if ignore the terms of higher than second order, weobtain corrected power spectrum as, P (2) φ ( k ) = ( H π ) (cid:16) − H Λ sin( 2Λ H ) + 12 ( H Λ ) −
14 ( H Λ ) sin( 2Λ H ) + ... (cid:17) , (35)This final result (34), includes linear and non-linearorder of trans-Planckian corrections H/ Λ. Martinand Brandenberger in equation (79) of paper [23] areobtained a similar set of corrections (29) and (34) up tosecond order of H/ Λ. V. CONCLUSIONS
In this paper, we have calculated higher order trans-Planckian corrections of power spectrum with excited-dS solution as the fundamental mode function during infla-tion. This consists essentially of expanding the Hankelfunction for the quantum mode in dS space to quadraticorder in kη before quantization, which corresponds toperforming the quantization at finite wavelength, ratherthan fully in the ultraviolet (i.e. Bunch Davies) limit.This non-trivial initial mode to be more logical sincethe curved space-time symmetry has been preserved af-ter renormalization procedure.In this excited- dS vacuum, slightly deviation of the ex-act solution lead to a correction to the power spectrumwhich is of non-linear order H/ Λ, where H is the Hub-ble parameter and Λ is the fundamental energy scale ofthe theory during inflation in very early universe. Thecorrections to the power spectrum that obtained withthis alternative mode is more complete than the correc-tions obtained from conventional method with pure dSmode, that of course in the dS limit leads to standardresult. Finally, due to selection of this excited mode,one expects creation of particles and non-Gaussianity ofCMBR, which will study in the future works. Acknowledgments
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